Comment on “HgS and HgS/CdS Colloidal Quantum Dots with Infrared Intraband Transitions and Emergence of a Surface Plasmon”

I ref 1, a model describing the surface plasmon resonances of semiconductor nanocrystals possessing extra band-like charge carriers is presented and contrasted with the model previously applied in ref 2. The latter is portrayed as having significant deficiencies that lead to erroneous interpretations. The two models were not adequately compared in ref 1, however, or in the work of ref 3 that is highlighted in ref 1 (and which was also published after ref 2). Upon comparison, we find that the two models are in fact equivalent, contain exactly the same physics, and have consequently led to the same conclusions about the experimental phenomenon under discussion. To determine the nanocrystal’s dipolar localized surface plasmon resonances (LSPRs), ref 2 considered the poles of the induced Green’s function associated with the quasistatic solution of Maxwell’s equations for a dielectric sphere. Contained within the sphere are many noninteracting quantum-mechanical electrons bound by an infinite sphericalwell potential of the same radius. Equivalently, ref 1 considered the poles of the induced polarization for the same problem but in the case of just one electron. Note that the induced polarization is derived from the induced Green’s function and shares the same poles; e.g., the sphere’s dipolar LSPR occurs at the frequency where ε(ω) + 2εm = 0. What is different between refs 1 and 2 is how these poles are calculated in practice. This is not a dif ference in model but rather a dif ference in implementation. Because of the mathematical simplicity of the one-electron case, ref 1 applied the condition ε(ω) + 2εm = 0 analytically exactly. This condition is equivalent to Re ε(ω) + 2εm = 0 together with Im ε(ω) = 0. In ref 2, many resonances were found to be significant, and the approximation Re ε(ω) + 2εm = 0 together with Im ε(ω) ≈ 0 was used. This well-established approximation has been invoked previously on numerous occasions; refs 4−7 represent just a few. This approximation was invoked because the exact high-order polynomial defining the nanocrystal’s dipolar LSPR was impossible to factor analytically. Instead, its roots were determined numerically using a graphical method to find the frequencies where Re ε(ω) crossed −2εm while simultaneously requiring Im ε(ω) ≈ 0. The only difference between the two implementations thus lies in the practical significance of this approximation. Analysis shows that the difference between using Im ε(ω) ≈ 0 and the exact Im ε(ω) = 0 results in variations of ≲1% over the full range of modeling results reported in ref 2. In light of the many idealizations and approximations inherent to the theoretical implementations of both papers, many of which were already discussed in ref 2, such a small discrepancy is not significant, and consequently it changes none of the conclusions drawn in ref 2. Ref 1 also raises the following issues with the way in which the ZnO dipolar LSPR frequencies are evaluated in ref 2: (1) the above approximation introduces an unphysical oscillatorstrength threshold that does not allow a smooth evolution from independent to collective electronic motion, (2) the above approximation generates a root in the region of anomalous dispersion, and (3) the use of the term “quantum plasmon.” Each of these points is now addressed: (1) The approximate way in which the roots of the polynomial Re ε(ω) + 2εm = 0 are determined does lead to an oscillator-strength threshold that must be overcome for a single(or many-degenerate) electronic transition(s) to contribute to collective motion, although not the threshold reported in ref 1. Equation 6 in ref 1 suffers from a typographical error that would indeed lead to drastic physical consequences if correct. Instead, the threshold is